TimeDomain CVD, Inc.

A Simple Reactor: Injector + Plug Flow

Orthogonal Fluxes:

In real systems both convection and diffusion are occurring simultaneously. Let's look at a very simple case.

A reactant is injected through a central passage (a slot in this case), while inert gases are dispensed near the walls. Here we've assumed that only the bottom wall has deposition, due perhaps to it being hot. How does the reactant get to the walls to react? How long does it take? Can we say anything about the concentration of reactant near the walls, and the thickness of the deposited film as a function of position (that is, can we understand the practically important parameters of deposition rate and uniformity)?

Plug Flow Reactor Model

 

To treat this problem we first need some nomenclature.   We will assume that the gas velocity is the same everywhere. (Recall that in a real case, you would know the molar gas flow, and could estimate the velocity by finding the volumetric flow from the ideal gas law, and then dividing by the known cross-sectional area, HZ).
We then make a key approximation, commonly employed: transport ALONG the streamlines is by convection only, and transport ACROSS the streamlines is by diffusion only. Since the velocity is uniform everywhere, transport along the streamlines is equivalent to letting the position be related to time by x = Ut. Thus, we can make use of the same diffusion equation as before to describe transport of species in the y direction, with the streamtime transformation t => x/U.
A solution that matches the boundary conditions at the entry to the reactor (x=0) can be constructed from our previous semi-infinite solutions. This solution remains approximately valid until the diffusion length grows long enough that the distribution has "reached" the walls (shown in the diagram at Ld = H/3).

Once the diffusion front reaches the reactor walls, what happens depends on the shape and relative rates of diffusion, convection, and reaction. We've already discussed the case where diffusion dominates (in that case, the inflection length x = H^2 * U/(36D) is very short compared to the chamber size). The two other simple cases are 1) diffusion is fast in the y-direction, but convection dominates in the x-direction ("simple depletion") or 2) diffusion is slow in the y-direction, and significant gradients in concentration persist in the y-direction.

Simple Depletion

We treat the downstream region as long and thin: diffusion in the y-direction is so rapid that the concentration is constant with respect to y, but changes in the x-direction. The net flux into a control volume arises from the derivative in x of the concentration, since the velocity is fixed. We assume deposition only occurs on the bottom surface; this is equivalent to a region with deposition on both surfaces but of height 2H.

 

As before, first we balance the fluxes entering and leaving a control volume:
leading to the differential equation:
with solutions of the form:

It is interesting here to recall our study of the zero-dimensional, or "continuously stirred" reactor. H is the ratio of the volume of a segment of the reactor to the surface area. Thus

the characteristic time for consumption of reactants. The depletion length is then the product of the consumption time and the velocity: that is, the solution is that of a zero-dimensional reactor moving at the fluid velocity U.

The final case is when diffusion in the y-direction is slow compared to the reaction. We leave this case as an exercise for the student (you should obtain a partial differential equation) and summarize the possibilities for depletion of a stream below.

 

Diffusion dominates convection: "Thiele" transport
Convection dominates in x-direction, diffusion is rapid in the y-direction: simple depletion
Convection dominates in the x-direction; reaction at the surface is rapid, leading to gradients in the y-direction

It is often possible to deal with rather complex geometries and situations in a similar fashion, by piecing together simple solutions (at least when Re is small!). The characteristic behavior exhibited in each regime is a useful guideline in more complex geometries where no analytic solution is possible.